 R. COLLINS21
compiled for the Survey of Occupational Injuries and
Illnesses (SOII) using methods described in Chapter 9
of the BLS Handbook of Methods .
It should be noted that these annual summaries are the
result of statistical analyses of raw data collected by BLS
from representative businesses within the industries
restricted. Moreover, the raw data collected by BLS is
only a sample collected to represent all businesses/estab-
lishments in the United States. Therefore, the data sum-
maries provided by BLS are statistical summaries of
samples collected in a manner so as to be representative
of all businesses/establishments in the United States.
know the shape of the data’s distribution or the distribu-
tion of data for all businesses/establishments in the
United States. Without knowledge of the shape of the
underlying data, we cannot know whether to use para-
metric or nonparametric statistical methods to analyze
the data in these summaries. Fortunately, the Central
Limit Theorem allows to assume that the sample means
and confidence limits obtained from summaries of the
underlying raw data are normally distributed because the
sample size is so very large. Had the underlying raw data
been available for analyses, the work presented in this
article would have benefited greatly. However, the
amount and variety of summary data available from the
underlying raw data is sufficient to allow us to perform
useful statistical analyses.
2.2. Data Management
Trends during 1992-2009 were analyzed to determine the
annualized and overall ratios of recordable incidents and
lost workday cases to fatal occupational injuries. This
data was used to determine the probability of experienc-
ing a fatal occupational injury per incident resulting in an
OSHA recordable incident for the time interval analyzed.
If we assume that accident prevention programs do not
change substantially over the next few years, these
probabilities derived from past incidents can be used to
predict future trends. With this assumption, probabili-
ties derived from these past incidents are used with gene-
rally accepted statistical methods to estimate the future
likelihood of fatal accident occurrence.
2.3. Data Analysis
The statistical methods chosen for this analysis are those
derived from Binomial distributions and Poisson distri-
butions. Binomial distribution methods are used because
they represent 1) a series of n independent events or tri-
als, 2) each of which can have only two possible out-
comes. In this case, a chronological series of OSHA re-
cordable incidents is the series of n independent events
or trials used. “Two events are statistically independent if
the probability of their occurring jointly equals the pro-
duct of their respective probabilities (i.e., Pr(AB) =
Pr(A) X Pr(B)).”  Each independent event is an inci-
dent resulting in an OSHA recordable incident that can
either result in a fatal or non-fatal injury. As can be seen,
a Binomial distribution is suitable for this analysis.
For recordable incidents alone, there are only two pos-
sibilities going forward. These are:
1) The recordable incident will result in a fatality; or
2) The recordable incident will not result in a fatality.
These are mutually exclusive and independent events,
otherwise known as a Bernoulli Variable. A toss of a
coin is a simpler example of a Bernoulli Variable. In
each case, the event can take on only 1 of 2 possible
values. Each toss of the coin or, in this case, recordable
incident represents a single Bernoulli Trial producing a
single result. When looking at a string of successive
Bernoulli Trials, the appropriate statistical method to use
in predicting potential future events of this type is a Bi-
nomial distribution. If we wanted to calculate the pro-
bability of having n successive trials before encountering
z specified outcomes, we would use the following gene-
ralized formula:


!
Probability 1
!!
nz
z
npp
zn z





where: n = Total number of consecutive trials or, in this
case, recordable incidents without a single fatality;
z = Total number of successes (or failures) or, in this
case, one or more fatalities;
p = Probability of a single success or failure (i.e., the
probability of one or more fatalities given n consecutive
recordable incidents).
Unfortunately, analysis using a Binomial distribution
alone gets us only part of the distance to a true temporal
probability relationship. In order to establish a temporal
probability relationship, a Poisson distribution must be
used. Poisson distributions are usually associated with
rare events. In this case, the rare event is a fatal occupa-
tional incident. Fortunately, Poisson distributions can be
used to approximate a Binomial distribution, under cer-
tain conditions. Generally accepted statistics theory al-
lows the use of a Poisson distribution to approximate a
Binomial distribution when the value of n (i.e., total
number of consecutive trials) is large and the value of p
(i.e., the probability of a single success or failure) is
small. In this case, the value of p is certainly small (i.e., p
= 1/987 = 1.01 × 103 ± 8.14%) and the value of n in-
creases with time. This means that the approximation of
a Binomial distribution using a Poisson distribution im- R. COLLINS
22
calculate. proves with time.
In this case, we will use the Binomial distribution
formula presented above to determine the probability of
having n successive OSHA recordable incidents without
a single fatality. In order to determine this probability,
we must first determine the probability of having n suc-
cessive OSHA recordable incidents without a single
success or failure (i.e., zero fatalities) using the single
event probability provided above (i.e., p = 1/987 = 1.01 ×
103 ± 8.14%). This simplifies the formula as follows:
Probability using a Poisson distribution can be calcu-
lated using the following formula:
e
Probability !
k
k
where: µ = Mean of the Poisson distribution = λt;
Λ = Number of events per unit time or number of fa-
talities per unit time;
t = Selected time interval;
k = Total number of successes (or failures) or, in this
case, one or more fatalities.



00
3
zero fatalities
Probability1.01100.999 n

Microsoft Excel and Stata/IC software (version 11,
StataCorp LP, College Station, Texas, USA) were used
for all of the statistical data analyses presented in this
paper. An alpha of p < 0.05 was selected as the level of
significance.
As this is the probability of having zero fatalities in n
successive OSHA recordable incidents, the probability of
having one (1) or more fatalities in n successive OSHA
recordable incidents can be determined using the fol-
lowing formula:



one (1) or more fatalities
00
3
zero fatalities
Probability
1Probability11.01100.999 n
 
3. Results
BLS data for all private industry has been extracted and
summarized in Table 1 from on-line summary tables.
When this data is normalized to the number of fatalities,
Figure 1 was created to show the ratio of recordable
incidents and lost workday cases to fatalities along with
the average values over the 18-year period shown. The
means and 95% confidence intervals for these ratios are
shown in Table 2. Over the 18-year interval studied,
there were an average of 987 recordable incidents and
476 lost workday cases for each fatality. If we assume
these ratios to be predictive as well as historical, we can
use them to determine the probability of future private
industry fatalities. The 95% confidence intervals (as
percents of the mean, shown in Table 2) can be consi-
dered to be error associated with the probabilities we
Collectively, these formulas produce the graph shown
in Figure 2. The x axis in Figure 2 represents the num-
ber of successive OSHA recordable incidents. In this
graph, the solid line represents the probability of having
n successive OSHA recordable incidents without having
a single fatality (i.e., zero fatalities). The dashed line is
the statistical complement to this (i.e., 1 – Probabilityzero
fatalities) showing the probability of having 1 or more fa-
talities in n successive OSHA recordable incidents. The
lighter lines above and below both of these lines repre-
sent the upper and lower bounds, given the error pro-
duced at the 95% confidence level (i.e.,
= 0.05). As
can be seen, the probability of having n successive
OSHA recordable incidents without a single fatality
Table 2. Ratios for all private industry and select industry groups.
95% Confidence Limits
Mean Lower Limit Upper Limit % Error
All Private Industry:
Ratio of OSHA Recordable Inc i dents to Fatalities 987 906 1,067 ± 8.14
Ratio of OSHA Lost Workday Cases to Fatalities 476 451 502 ± 5.27
NAICS Code 238 (Specialty Trade Contractors Industry Group):
Ratio of OSHA Recordable Incidents to Fatalities 442 407 477 ± 7.83
Ratio of OSHA Lost Workday Cases to Fatalities 221 209 232 ± 5.11
NAICS Code 211 (Oil and Gas Extraction Industry Group):
Ratio of OSHA Recordable Incidents to Fatalities 167 142 191 ± 14.4
Ratio of OSHA Lost Workday Cases to Fatalities 93 76 109 ± 17.4 R. COLLINS23
Figure 2. Fatality probabilities per consecutive recordable incident. Upper and lower confience limits (i.e., % error) are
shown as narrower lines above and below.
drops significantly while the probability of having one or
more fatalities in n successive OSHA recordable inci-
dents climbs exponentially approaching 1 (or 100%)
asymptotically.
In order to use a Poisson distribution to approximate
the Binomial distribution results, we have all of these
values in the formula provided above except for the
value of λ (i.e., the number of events per unit time). We
do have the value of p (i.e., the probability of a single
success or failure per recordable incident), however. We
can use the value of p to get the value of λ if we know
the temporal frequency of recordable incidents. The
formula for determining the value of λ from the value of
p, given the temporal frequency of recordable incidents
(f), is:
pf

where: λ = Number of events per unit time
p = Probability of a single success or failure per re-
cordable incident
f = Temporal frequency of recordable incidents per
unit of time
The astute reader will recognize that the temporal fre-
quency of recordable incidents is directly related to size
of the business (i.e., the number of hours worked each
year). It may also be a function of the less quantifiable
variables of 1) intrinsic risk of the business; and 2) qua-
lity of the accident prevention program.
For the purposes of this example, let’s assume that the
value of f is 100 recordable incidents per year. Substi-
tuting into the formula above, we get a value for λ equal
to:


3
1
1.01 10Fatalities100Recordable Incidents
Recordable IncidentYear
Fatalities
1.0110Year


Substituting this value for λ into the probability func-
tion above yields the following results:


1
1.01 10fatalitiesYear1
Probability
e1.0110fatalities Year
!
k
tt
k
 
 
Figure 3(a) shows the results of this equation graphi-
cally. In this case, the x axis is time in years. The solid
line represents the probability of having zero (0) fatali-
ties (i.e. Probabilityzero fatalities) in the time intervals indi-
cated in years. The dashed line is the statistical comple-
ment to this (i.e., 1 – Probabilityzero fatalities) showing the
probability of having 1 or more fatalities (i.e., Probabili-
tyone (1) or more fatalities) in the time intervals indicated in
years. The lighter lines above and below both of these
lines represent the upper and lower bounds, given the
error produced at the 95% confidence level (i.e.,
=
0.05). As can be seen, the probability of having zero (0) R. COLLINS
24
fatalities drops significantly while the probability of
having one or more fatalities in the time intervals indi-
cated in years climbs exponentially approaching 1 (or
100%) asymptotically. For this example of 100 record-
able incidents per year, the probability of having one or
more fatalities exceeds 50% after only 7 years.
If we were to use a different example, say 10 record-
able incidents per year, the graph (Figure 3(b)) would
appear the same but the x axis would reach further into
the future. In this example, when the x axis is expanded
to 500 years the graph is very similar in appearance. For
10 recordable incidents per year, the probability of hav-
ing one or more fatalities exceeds 50% after only 69
years.
4. Discussion
It should be noted that the ratios of 477 lost workday
cases per fatality and 987 recordable incidents per fatal-
ity (i.e., 1:477:987) were derived for all private industry.
In reality, private industry within the United States is
(a)
(b)
Figure 3. Fatal probability per year for all Private Industry. Solid line indicates the probability of zero (0) fatalities per year
while the dashed line indicates the probability of one (1) or more fatalities per year. Lighter shaded lines above and below
represent the upper and lower confidence limits (i.e., % error). Figure 3(a) assumes a rate of 100 recordable incidents per
ear. Figure 3(b) assumes a rate of 10 recordable incidents per year. y R. COLLINS
25
types and individual businesses would have the same risk
profile. We can see this by looking at two specific exam-
ples. In the most recent year for which records are avail-
able (i.e., 2009), Specialty Trade Contractors (North
American Industrial Classification System or NAICS
Code 238) had the highest total number of fatalities (i.e.,
477) while Oil and Gas Exploration firms (NAICS Code
211) had among the lowest total number of fatalities (i.e.,
17). When we apply the same methodology used above
to characterize risk within private industry, we get the
risk profiles shown in Table 2. From these results, it can
be seen that the risk profile for private industry (i.e.,
1:476:987) is significantly different from the risk profiles
for Specialty Trade Contractors (i.e., 1:221:442) and Oil
and Gas Exploration (i.e., 1:93:167). These risk profiles
have nothing to do with the size of the business (i.e., the
number of hours worked each year) and everything to do
with those less quantifiable variables of 1) the intrinsic
risk of their businesses; and/or 2) the quality of their ac-
cident prevention programs.
In any event, different risk profiles mean different
probabilities for a single success or failure (i.e., p) that
can be applied as above. The resulting probability graphs
(Figure 4(b) for Specialty Trade Contractors and Figure
4(c) for Oil and Gas Exploration), assuming 100 record-
able incidents per year, appear very similar to that for all
private industry (i.e., Figure 4(a)) with only a shift to the
right or left on the x axis. However, as can be seen, the
probability of having one or more fatalities exceeds 50%
after 3 years for Specialty Trade Contractors (NAICS
Code 238) and 1 year for Oil and Gas Extraction
(NAICS Code 211), assuming an annual rate of 100 re-
cordable incidents per year. This would seem to indicate
that working in the Oil and Gas Extraction industry
group (NAICS Code 211) is more dangerous (at least
from the likelihood of death following an incident) than
it is in the Specialty Trade Contractors industry group
(NAICS Code 238) and working in either industry group
appears to be more dangerous than working in private
industry as a whole. This is an especially interesting re-
sult, given that two industry groups with diametrically
opposite fatality totals during 2009 (i.e., 17 and 477)
were chosen. This may be explained by the fact that,
based upon hours worked, the Oil and Gas Extraction
industry group (NAICS Code 211) is 1/10th of the size of
the Specialty Trade Contractors industry group (NAICS
Code 238) and this group is 1/100th of the size of all pri-
vate industry.
This same type of refined analysis can be applied to an
individual business/establishment if there is sufficient
historical data on which to base a potential future prob-
ability. Figure 5 shows how this might be applied to a
single business/establishment. The x axis in this figure
represents years while the y axis represents probability.
The solid line descending from left to right represents the
probability of experiencing zero (0) fatal incidents for
the general industry type corresponding to a specific
business/establishment while the dashed line descending
from left to right represents the same probability for a
specific business/establishment. The solid line ascending
from left to right represents the probability of one (1) or
more fatal incidents for the general industry type corre-
sponding to a specific business/establishment while the
dashed line descending from left to right represents the
same probability for a specific business/establishment.
The vertical dashed line represents the current position in
time on the graph for this specific business/establishment.
Two (2) potentially valuable lessons that can be learned
from this figure are:
1) The dashed lines representing a specific business/
establishment are to the left of the solid lines represent-
ing the general industry type to which the specific busi-
ness/establishment belongs. This can be interpreted as
meaning that it is more dangerous to work in this specific
business/establishment than it is for the group to which
2) At the specific point in time indicated by the verti-
cal dashed line, the probability that the next OSHA re-
cordable incident will result in a fatal accident is 35.6%
for the specific business/establishment while it would
only be 19.9% for the group to which this business/es-
tablishment belongs.
It should be noted, however, that an individual busi-
ness/establishment will be a subset of the industry group
to which it belongs. Moreover, both the Oil and Gas Ex-
traction industry group (NAICS Code 211) and the Spe-
cialty Trade Contractors industry group (NAICS Code
238) are subsets of the larger private industry group.
While restricting this analysis to a specific industry
group or business/establishment might produce more
useful results, the confidence interval size (i.e., error)
must of necessity increase as well making the results
potentially less reliable. Statistically speaking, confi-
dence intervals tend to increase as the sample size de-
creases. In practice, however, two (2) different busi-
nesses within the same industry group may have widely
different approaches to accident prevention. A business
with limited efforts to prevent accidents, sometimes
known as a poor safety culture, may find their accident
prediction curve (i.e., the Poisson distribution demon-
strated above) shifted to the left compared to the industry
group. Whereas, a business with strong efforts to prevent
accidents, sometimes known as an excellent safety cul-
ture, may find their accident prediction curve shifted to
the right. In this example, smller sample sizes are more a R. COLLINS
26
(a)
(b)
(c)
Figure 4. Fatal probability per year, assuming a rate of 100 recordable incidents per year. Solid lines indicate the probability
of zero (0) fatalities per year while the dashed lines indicate the probability of one (1) or more fatalities per year. Lighter
shaded lines above and below represent the upper and lower confidence limits (i.e., % error). Figure 4(a) is for all Private
Industry. Figure 4(b) is for the Specialty Trade Contractors industry group (NAICS Code 238). Figure 4(c) is for the Oil and
Gas Extraction industry group (NAICS Code 211). R. COLLINS27
Figure 5. Fatal probability per year for a single business/establishment.
susceptible to variations in the quality of accident pre-
vention policies, programs and procedures.
Readers of Heinrich’s original work will note that the
three (3) levels of his original triangle consisted of 1)
Major Injuries; 2) Minor Injuries; and 3) No-Injury Ac-
cidents. The definitions provided for those original three
(3) categories were:
1) Major Injuries: “Any case that is reported to insu-
rance carriers or to the state compensation commis-
sioner” . In today’s vernacular, these would be OSHA
recordable cases.
2) Minor Injuries: “A scratch, bruise, or laceration
such as is commonly termed a first-aid case” .
3) No-Injury Accidents: “An unplanned event involve-
ing the movement of a person or an object, ray, or sub-
stance (slip, fall, flying object, inhalation, etc.), having
the probability of causing personal injury or property
damage.” .
From this we can see, that the three (3) levels of the
(i.e., fatalities, lost workday cases and OSHA recordable
cases) would all fit into only the top level of Heinrich’s
original triangle. The bottom two levels of Heinrich’s
original triangle would appear as two additional layers of
this New Incident Pyramid. From Heinrich’s original
triangle, we know only that these two (2) bottom levels
of the New Incident Pyramid would be located below the
OSHA recordable case level and that they would be lar-
ger than the OSHA recordable case level. Unfortunately,
there are no reliable resources to quantify these addi-
tional levels. If we were to use the ratios provided in
Heinrich’s original work, the New Incident Pyramid
would appear as seen in Figure 6 for all private industry.
If data for these two (2) additional levels of the New In-
cident Pyramid were more current and reliable, we could
use them in the same type of analyses presented in this
article to predict potential future fatalities based upon the
total number of all incidents instead of only OSHA re-
cordable incidents. As there are no accurate and reliable
modern day resources from which data can be gathered
for these additional analyses, we can rely only on the
5. Conclusions
It should be noted that the statistical analyses presented
rived from raw data collected by BLS from representa-
tive businesses. The statistical analyses performed within
the underlying raw data in lieu of these summary tables.
One of the fundamental underlying principals of
Heinrich’s original triangle is that fatalities cannot occur
without a foundation of less severe incidents. In other
words, increasing numbers of non-serious incidents
eventually support the existence of more serious and
fatal incidents. We can now see from the analyses pre-
sented in this paper that this also applies to incidents R. COLLINS
28
Figure 6. New incide nt pyramid for all private industry.
occurring over time. Valuable conclusions that can be
drawn from this analysis include:
1) Fatal accidents are inevitable. Each day that passes
without a fatal occupational accident only leads to an
increased likelihood that a fatal occupational accident
will occur during the next day as the probability ap-
proaches 100% asymptotically. This conclusion may be
dependent on the occurrence of less severe incidents as
well as recordable incidents, as demonstrated by the
adaption of Heinrich’s original triangle shown in Figure
6.
2) Factors potentially affecting when these fatalities
will occur include:
a) Size of the business (i.e., number of hours worked
each year);
b) Intrinsic risk of the business; and/or
c) Quality of the accident prevention program.
As only one (1) of these factors is within the direct
control of business managers, the likelihood of a poten-
tial future fatal accident can only be decreased by con-
tinually improving the quality of their accident preven-
tion programs.
Perhaps the central lesson to be learned from these
analyses is that we must always look on recordable inci-
dents (and even accidents without injuries) as portents of
fatal accidents to come. Doing little to prevent recordable
or minor accidents today will only lead to fatal incidents
sooner rather than later; if we do not respond appropri-
ately when they occur. It is only by considering safety
and accident prevention with our every thought and ac-
tion that we can hope to push these fatal accidents further
off into the future by shifting our own personal probabi-
lity distribution to the right.
6. References
 H. H. William, “Industrial Accident Prevention: A Scien-
tific Approach,”4th Edition, McGraw-Hill Book Com-
pany, Boston, 1959.
 R. L. Collins, “Heinrich and Beyond,” Process Safety
Progress, Vol. 30, No. 1, 2011, pp. 2-5.
 B. Rosner, “Fundamentals of Biostatistics,” Brooks/Cole, 